In the above definition, at each \(t\in (0,T)\) , \(u(t) = (\bar{u}(t), t)\)  where \(\bar{u}(t)\)  is an element of \(X(t)\)  for almost every \(t\) .
We will identify \(u(t)\)  with \(\bar{u}(t)\)  for convenience.
We note that our definition of \(L^2_{X^*}\)  does not use the map \(\phi _t^*\)  as appears in the corresponding definition of [1]}.
We are instead using the map \(\phi _t\)  as in Definition REF .
